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Properties

Label 17.1.18372639664...8561.1
Degree $17$
Signature $[1, 8]$
Discriminant $13^{8}\cdot 83^{8}$
Root discriminant $26.75$
Ramified primes $13, 83$
Class number $1$
Class group Trivial
Galois group $D_{17}$ (as 17T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![49, 35, 180, -62, 16, -4, 112, 30, -62, 39, -48, 44, -30, 26, -19, 5, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^17 - x^16 + 5*x^15 - 19*x^14 + 26*x^13 - 30*x^12 + 44*x^11 - 48*x^10 + 39*x^9 - 62*x^8 + 30*x^7 + 112*x^6 - 4*x^5 + 16*x^4 - 62*x^3 + 180*x^2 + 35*x + 49)
 
gp: K = bnfinit(x^17 - x^16 + 5*x^15 - 19*x^14 + 26*x^13 - 30*x^12 + 44*x^11 - 48*x^10 + 39*x^9 - 62*x^8 + 30*x^7 + 112*x^6 - 4*x^5 + 16*x^4 - 62*x^3 + 180*x^2 + 35*x + 49, 1)
 

Normalized defining polynomial

\( x^{17} - x^{16} + 5 x^{15} - 19 x^{14} + 26 x^{13} - 30 x^{12} + 44 x^{11} - 48 x^{10} + 39 x^{9} - 62 x^{8} + 30 x^{7} + 112 x^{6} - 4 x^{5} + 16 x^{4} - 62 x^{3} + 180 x^{2} + 35 x + 49 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $17$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1837263966425479287178561=13^{8}\cdot 83^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.75$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $13, 83$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{13} a^{11} - \frac{4}{13} a^{10} + \frac{3}{13} a^{9} + \frac{1}{13} a^{8} - \frac{6}{13} a^{7} - \frac{2}{13} a^{6} - \frac{5}{13} a^{5} - \frac{6}{13} a^{4} - \frac{4}{13} a^{3} + \frac{3}{13} a^{2} + \frac{1}{13} a - \frac{4}{13}$, $\frac{1}{143} a^{12} + \frac{1}{143} a^{11} + \frac{9}{143} a^{10} + \frac{29}{143} a^{9} - \frac{27}{143} a^{8} - \frac{6}{143} a^{7} + \frac{50}{143} a^{6} - \frac{70}{143} a^{5} - \frac{8}{143} a^{4} + \frac{35}{143} a^{3} - \frac{36}{143} a^{2} + \frac{53}{143} a + \frac{58}{143}$, $\frac{1}{143} a^{13} - \frac{3}{143} a^{11} + \frac{64}{143} a^{10} + \frac{54}{143} a^{9} + \frac{10}{143} a^{8} - \frac{21}{143} a^{7} + \frac{45}{143} a^{6} - \frac{2}{11} a^{5} - \frac{34}{143} a^{4} - \frac{27}{143} a^{3} + \frac{56}{143} a^{2} - \frac{6}{143} a - \frac{14}{143}$, $\frac{1}{143} a^{14} + \frac{1}{143} a^{11} + \frac{59}{143} a^{10} + \frac{42}{143} a^{9} - \frac{25}{143} a^{8} - \frac{6}{143} a^{7} - \frac{30}{143} a^{6} - \frac{57}{143} a^{5} + \frac{59}{143} a^{4} - \frac{4}{143} a^{3} - \frac{2}{11} a^{2} - \frac{64}{143} a + \frac{9}{143}$, $\frac{1}{1001} a^{15} + \frac{2}{1001} a^{14} - \frac{3}{1001} a^{13} - \frac{30}{1001} a^{11} + \frac{69}{1001} a^{10} + \frac{2}{7} a^{9} + \frac{271}{1001} a^{8} - \frac{380}{1001} a^{7} - \frac{30}{77} a^{6} + \frac{302}{1001} a^{5} - \frac{326}{1001} a^{4} - \frac{450}{1001} a^{3} - \frac{116}{1001} a^{2} - \frac{36}{91} a + \frac{16}{143}$, $\frac{1}{1221517297} a^{16} - \frac{36983}{1221517297} a^{15} - \frac{3908517}{1221517297} a^{14} + \frac{3036619}{1221517297} a^{13} + \frac{28187}{1221517297} a^{12} - \frac{4920906}{174502471} a^{11} + \frac{553749892}{1221517297} a^{10} - \frac{206982179}{1221517297} a^{9} - \frac{3703743}{28407379} a^{8} - \frac{30236515}{111047027} a^{7} - \frac{10006336}{174502471} a^{6} + \frac{147306151}{1221517297} a^{5} + \frac{33562096}{174502471} a^{4} + \frac{505944163}{1221517297} a^{3} + \frac{171129719}{1221517297} a^{2} - \frac{52795685}{1221517297} a - \frac{65347462}{174502471}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 257874.899874 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{17}$ (as 17T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 34
The 10 conjugacy class representatives for $D_{17}$
Character table for $D_{17}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Galois closure: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $17$ $17$ $17$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ R $17$ $17$ $17$ $17$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $17$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
$83$$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
83.2.1.1$x^{2} - 83$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.1$x^{2} - 83$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.1$x^{2} - 83$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.1$x^{2} - 83$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.1$x^{2} - 83$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.1$x^{2} - 83$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.1$x^{2} - 83$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.1$x^{2} - 83$$2$$1$$1$$C_2$$[\ ]_{2}$